Elastic Shaft Research Articles

A 3-node Shaft Element for Main Shaft Vibration FE Analysis

To calculate the characteristics of the lateral vibration in the main shaft system of the hydro-turbine generating set, a three-node elastic shaft element was proposed. Its dynamic equation was derived and the interpolation function, translational inertia matrix, rotational inertia matrix, gyroscopic matrix and the stiffness matrix were obtained. Based on Matlab platform, a FEM program was developed for calculation and its correctness was verified through a numerical example. The study shows that, compared with the method using two-node shaft element, the results of the three-node shaft element are closer to the theoretical solution and the accuracy is higher.

Analysis on Torsional Vibration Characteristics of Tracked Vehicle Gearbox

According to the dynamic characteristics of tracked vehicle gearbox system, the integrated influence of elasticity of shaft, gear meshing stiffness and transmission error in the system is considered, and mathematics models of two type configurations are created, Matlab/Simulink model of which are provided. The torsional vibration dynamic model of transmission system with tracked vehicle gearbox is developed and the vibration equation of the system is established. The influences of the transmission errors, moment of inertia, mesh phase and torsion stiffness are investigated respectively. It has important realistic meaning for guiding the gearbox structural design and improving, reducing its fault rate.

Analysis and Controller Design of a Torsional D.C. Drive under Resonance Conditions

The performance of an electric drive not only depends upon electrical parameters but is also significantly affected by mechanical features of drive like, elasticity of shaft, misalignment of shaft, eccentricity of driven load. For precise determination of performance, especially under dynamic conditions, the effect of these mechanical factors should be included in the analysis. The work reported in this paper deals with the analysis of a D.C. motor drive with elastic coupling and periodically varying load torque. Periodic variation of load torque may due to mechanical features or may be the characteristics of the driven load itself. A mathematical model of the drive system is developed and the system equations are expressed in state model form. Simulink models of the drive are developed and performance of drive system is studied for constant as well as pulsating load torque conditions. Under some specific conditions, system goes under resonance. Conditions leading to resonance are identified and drive performance under normal as well as resonance conditions is analyzed. The analyses presented reveal that mechanical failure of the drive system may occur due to excessive shear stress developed in the shaft under resonance conditions. Design of a controller is suggested to protect the drive from such failures under resonance condition.

Dynamics of a Flexible Rotor With Circumferentially Non-Uniform Magnetization Supported by a Superconducting Magnetic Bearing

A superconducting magnetic bearing has many opportunities in which it is driven higher than their critical speeds. Under those circumstances, the rotor can easily show elastic deformation and complicated phenomena by nonlinearity of electromagnetic force. Therefore, evaluation of its dynamics taking nonlinearity and elasticity into consideration in the vicinity of its critical speeds is demanded. This research examines the effects of magnetic nonuniformity of a rotor in its circumferential direction on dynamics. First, an essential model of a rotor with an elastic shaft supported by a superconducting magnetic bearing was introduced. By taking account of magnetic non-uniformity, our numerical result showed its three effects: anisotropy of the rigidity, characteristics equivalent to those of an asymmetrical cross section of the axis, and nonlinearity of electromagnetic force. These effects cause difference of critical speeds in the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$x$</tex></formula> , <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$y$</tex></formula> directions, a frequency band showing instability of the rotor's motion, and change of eigenfrequencies depending on the amplitude, respectively. These effects may emerge simultaneously when the rotor has magnetic nonuniformity in its circumferential direction.

Finite Element Multibody Simulation of a Breathing Crack in a Rotor with a Cohesive Zone Model

The breathing mechanism of a transversely cracked shaft and its influence on a rotor system that appears due to shaft weight and inertia forces is studied. The presence of a crack reduces the stiffness of the rotor system and introduces a stiffness variation during the revolution of the shaft. Here, 3D finite element (FE) model and multibody simulation (MBS) are introduced to predict and to analyse the breathing mechanism on a transverse cracked shaft. It is based on a cohesive zone model (CZM) instead of linear-elastic fracture mechanics (LEFM). First, the elastic cracked shaft is modelled by 3D FE. As a second step, the 3D FE model of the shaft is transferred into an MBS model in order to analyze the dynamic loads, due to the crack, and the inertia force acting during rotation at different rotating speeds. Finally, the vibration responses in the centroid of the shaft obtained from MBS have been exported into FE model in order to observe the breathing mechanism. A bilinear crack closure model is proposed. The accuracy of the bilinear crack closure model and the solution techniques have been demonstrated by a comparison with the corresponding results of previous publications.

Load-side Sensor-less Speed Regulation of a Two-Inertia System with an Eccentricity

Motor drive systems are widely used in the industrial field. The rotor stiffness turns down relatively due to the demand of weight saving, and so on. Then the torsional vibration in the drive system appears because of the elasticity of shaft. Such systems consisting a motor and a load with elastic shaft often are modeled as the two-inertia system. When the eccentricity exists in the load, the system has difficulty to follow the constant load speed reference. This paper proposes both an identification method of the eccentricity in the load and a control method to regulate the rotating speed of load without using load-side sensors when the eccentricity exists.

The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems

It is shown that the effect of dissipative and gyroscopic forces on a certain class of potential linear time-varying system differs considerably from the effect of these forces on a time-invariant system. Examples are considered. In the first of these, the equations of motion of a disk, attached to a rotating weightless elastic shaft, are investigated, taking external friction into account. The results obtained are compared with the results obtained previously by others when considering this problem. In the second example, certain problems of the stability of rotation of a Lagrange top on a base subjected to vertical harmonic vibrations are investigated.

Auto-balancing of a rotor with an orthotropic elastic shaft

The self-balancing of a statically unbalanced orthotropic elastic rotor equipped with a ball auto-balancing device is investigated. Equations of motion in fixed and rotating systems of coordinates, as well as equations describing steady motions of the regular precession type, are derived using a simple model of a Jeffcott rotor. Formulae for calculating the amplitude-frequency and phase-frequency characteristics of the precessional motion of the rotor are obtained. It is established that the conditions for a steady balanced mode of motion for an orthotropic rotor to exist have the same form as for an isotropic rotor, but the stability region of such a mode for an orthotropic rotor is narrower than the stability region for an isotropic rotor. The unsteady modes of motion of the rotor in the case of rotation with constant angular velocity and in the case of passage through critical velocities with constant angular acceleration is investigated numerically. It is established that the mode of slow passage through the critical region for an orthotropic rotor is far more dangerous than the similar mode for an isotropic rotor.

Instability of a general rotating system with small axial asymmetry and damping

The paper presents new results on instability of general rotating systems with small axial asymmetry and damping. Rotating systems with arbitrary finite degrees of freedom are considered. Instability conditions are derived for nonresonant and resonant rotation speeds characterized by simple and double eigenvalues. As an application, the stability problem of a rotating elastic shaft with small internal and external damping forces and asymmetric imperfections is studied. The numerical example shows validity and accuracy of the derived analytical formulae for the instability regions.

Internal resonance of a flexible rotor supported by a magnetic bearing

Magnetic bearings can support rotating rotors without any contacts. So itispossible torotatea rotor at extremely high rotational speed. On the other hand, they can easily show elastic deformation with nonlinear oscillation, which is complicated and difficult to deal with. This study investigates nonlinear coupling of oscillation modes appearing in such a flexible rotor system with a magnetic bearing, by considering both the nonlinearity of the electromagnetic force and the continuousness of the rotor. First, an essential model of a rotor with an elastic shaft supported by a magnetic bearing was introduced. The numerical results indicate occurrence of internal resonance caused by the nonlinear coupling of the modes, if the ratio of their natural frequencies is 1 to 3 Experiments were also carried out and obtained results showed qualitative accordance to the numerical results.

Influence of Centrifugal Force on Characteristics of Turbine Shaft for Micro-Spindle for Micro-Cutting

The centrifugal force has great influence on characteristics of the turbine shaft. The changes of stress, diameter and natural frequency of the turbine shaft for self-developed micro-spindle for micro-cutting following with the rotational speed were studied by finite element (FE) simulation, which showed that stress, diameter of the turbine shaft and frequency of torsional vibration in elastic shaft coupling integrated with the turbine shaft caused by the centrifugal force increases almost linearly with increase of rotational speed and the higher the speed, the larger the increase. The frequencies of other vibrational modes were influenced by centrifugal force. Materials with good properties can improve characteristics of turbine shaft caused by the centrifugal force well.

Resonances and Aerodynamic Damping of a Vertical Axis Wind Turbine

The dynamics of a straight-bladed vertical axis wind turbine is investigated with respect to oscillations due to the elasticity of struts and shaft connecting to the hub. In particular, for the three-bladed turbine, a concept is proposed for dimensioning the turbine to maximize the size of the resonance free rpm range for operation. The effect of aerodynamic damping on the struts is also considered. The damping of these types of oscillations for a typical turbine is found to be good.

Evaluation of elastic shaft dumping ratio for the research of dynamic processes in transmission

A method of evaluation of elastic shafts dumping ratio is presented. It involves a procedure of preliminary calculation of dumping ratio according to the mass and geometric characteristics of the shaft and the damping coefficient of shaft material, as well as a procedure of experimental correction of previously calculated values. The introduced approach allows to find the dumping ratios of transmission elastic shafts with sufficiently high precision.

Optimal Design of Rotation Shaft Torque Testing System

Torque measuring in the extra-high rotation speed has not been resolved ideally yet, due to the restriction of the rough conditions. The dynamic characteristics of the developed digital plane grating angular-torquer measuring system are analyzed in this paper. The paper mainly calculates and designs the parameters of elastic shafts with different cross-section. Via the compare of dynamic characteristics and compactness of the elastic shafts with different cross-section, the optimized design is proposed at the end of the paper. The achievements of this paper can supply references to the study and design, maintaining and improvement of relative instruments.

Load Allocation and Equilibrium for Planetary Gear Reducers of Earth-Pressure-Balance Shield Machine

To optimize the 3-stages planetary gear reducer (PGR) in the Earth-Pressure-Balance Shield Machine (EPBSM), it is necessary to firstly define the EPBSM environment. The needed torque of the cutter-head, therefore, is analyzed when the EPBSM cutting the earth body. The load of any PGR is allocated according to the equivalent principle in the size between the force and its response one and the distributing uniformity of the PGRs around the main axle gear. Because the structure of the main drive system is equal to a huge planet-gear reducer made of 8 planetary gears, for the main axle connected with it gear, one side is connected with a bidentate coupling, the other is connected with an elastic shaft to realize the dual connection floatability of the main axle gear and support the whole main axle gear so that the relative displacement can be compensated to high degree among the PGRs resulting from the uneven force when the cutter-head cuts earth body.

The stability of a non-potentially loaded discrete-continuous gyroscopic system with internal and external friction

The problem of the stability of the axial rotations of a symmetric rotor by means of an elastic shaft in the form of a thin straight rod, when there is a follower force and linear dissipation, is considered. Linearized equations for the displacement of the rotation of the rotor and the transverse plane oscillations of the rod are presented in a complex form as well as the boundary conditions. Using the standard procedure of a Laplace transformation with respect to time, relations for the images are obtained which are then solved by means of quasipolynomials of the complex transform parameter, and an asymptotic analysis of them is carried out. The effect of a follower force is investigated and its critical values are determined. The results of mathematical modelling for specific numerical values of the parameters are presented.

Nonlinear Dynamics of Jeffcott Rotor System with Rub-Impact Fault

The rub-impact fault of Jeffcott rotor system which is generally made of an elastic shaft with supported by two bearings at ends and a lumped mass at the mid-span of shaft is the main research subject of this paper. The mechanical model and finite element model of a lump mass at the mid-span of shaft is set up. The dynamics of it are studied by nonlinear finite element method under the rub-impact conditions, and by investigating the effects of stiffness of stator and the clearance between rotor-stator. From the study results, use finite element method to study the rotor system failure problem is intuitive and convenient, high accuracy, high reliability, can be widely applied.

Bending Stiffness Analysis of Elastic Shafts with Three Types of Transverse Cracks

The breathing mechanism of a transverse crack in an elastic shaft has been extensively researched. Although many different crack models have been presented, little published research has explored experimentally how the shaft crack breathing mechanism interacts with the change of stiffness during each revolution. This study addressed static analysis of the stiffness of elastic shafts with experimentally-induced fatigue, welding or wire cut transverse cracks. Bending stiffness was measured from different angles under a three point bending fixture, to efficiently define the correct periodical stiffness variation for different crack types. The opening and closing of each crack (breathing effect) were investigated for their association with shaft stiffness. It was found that width of the crack opening gap is a crucial factor which influences the crack breathing behaviour. Experimental results are expected to assist with crack diagnosis.

Dynamic Analysis and Research of Motor’s Elastic Shaft of the Large Miner High Shearer’s Cutting Parts

Take one large miner high electric draught shearer for example. According to the lumped mass method, the gear’s moment of inertia of the shearer’s cutting part was equivalent to the gears connecting the output shaft of the electric cutting motor, the stiffness of central shaft and coupling was equivalent to the output elastic shaft of the electric cutting motor, and the dynamic analysis model of the elastic shaft of the cutting part was set up. Based on this model, the lagrange equation was built. The critical speed of the elastic shaft system, the motion situation on main vibration and harmonic vibration mode, motor shaft’s angular displacement, angular velocity, and angular acceleration’s dynamic response process under the condition of step loads were all analyzed and obtained. The critical speed provided theoretical basis for choosing the working speed of every shaft system of the shearer. The hollow elastic shaft had a protective effect to cutting motor’s damping and overload. The research results provided theoretical basis for the design of the elastic shaft of large miner high shearer.

Experiment and Test Signal Analysis Based on Speed Fluctuation of Torsional Vibration of Large-Scale Rotary Machinery

A new mode of fault monitoring and controling methods on rotating speed fluctuation was proposed in this study. Torsional vibration model and identification equation of speed fluctuation of large-scale rotary machinery was established based on a two-mass motor driving model. Tachogenerator was adopted to measure speed fluctuation in torsional vibration experiment of large-scale rotary machinery. According to the short time fourier transforms method, the non-steady cyclical or quasi-cyclical characteristics signal of rotating speed fluctuation on elastic shafts were transformed into steady signal to study in a fixed time window function. The methods of monitoring rotating speed fluctuation developed nonlinear stable state signal processing into linear short time fourier transforms signal. The real rotating speed fluctuation solution could be obtained after the data of signal acquisition post-processing by the methods of frequency spectrum analysis and modal analysis. Based on data of signal acquisition, using the methods of fourier phase frequency spectrum, logarithm amplitude frequency spectrum, and self-power spectrum, the quantitative expression under the quantitative analysis stable state was obtained. Through the introduction of realtime signal on rotating speed fluctuation to feedback control system, it is easy to program to realize the real-time on-line torsional vibration monitor of the complex mechanism transmission system on the large scale rotating machinery.